Parametric Equations — Challenge
Let a third variable drive both coordinates, and curves that were impossible become easy.
What these problems cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Radius — Centre to edge. Half the diameter.
- Slope — How much y changes for each 1 that x increases.
- Parabola — The U-shaped graph of a quadratic function.
- Roots — The x values where y = 0.
- Cosine — cos θ = adjacenthypotenuse.
- Pythagorean identity — sin2θ + cos2θ = 1.
- Continuous — A function you could draw without lifting the pen — no holes, jumps or blow-ups.
Problem 1
Eliminate the parameter from x = t2, y = t3.
- y2 = x3
- y = x3/2 only
- y3 = x2
- y = x
- x = y
Worked answer
y2 = x3
Cubing the first gives x3 = t6; squaring the second gives y2 = t6. Both equal t6, so y2 = x3. Writing y = x3/2 would lose the lower half of the curve, since t may be negative while x = t2 stays positive.
Problem 2
Eliminate the parameter from x = cos t, y = cos 2t.
- y = 2x2 - 1
- y = x2
- y = 2x2 + 1
- y = x2 - 1
- y = 2x
Worked answer
y = 2x2 - 1
The double-angle identity gives cos 2t = 2cos2 t - 1. Since x = cos t, that is y = 2x2 - 1. The curve is only the part of that parabola with -1 ≤ x ≤ 1, since a cosine cannot leave that range.
Problem 3
A projectile has x = 30t and y = 40t - 16t2. How far does it travel horizontally before landing?
- 75
- 30
- 100
- 62.5
- 40
Worked answer
75
It lands when y = 0: t(40 - 16t) = 0, so t = 2.5 seconds. Substitute into the horizontal equation: x = 30(2.5). = 75. Horizontal motion is steady, so no gravity term appears there.
Problem 4
Eliminate the parameter from x = 2+3t, y = 1-t.
- x + 3y = 5
- 3x + y = 5
- x - 3y = 5
- x + y = 3
- 3x - y = 7
Worked answer
x + 3y = 5
From the second equation, t = 1 - y. Substitute: x = 2 + 3(1-y) = 5 - 3y. Rearranging gives x + 3y = 5 — a straight line, as two linear parametric equations must produce.
Problem 5
What curve does x = 3cos t, y = 2sin t trace?
- An ellipse 3 wide and 2 tall
- A circle of radius 3
- A circle of radius 2
- A hyperbola
- A straight line
Worked answer
An ellipse 3 wide and 2 tall
Rearrange: x3 = cos t and y2 = sin t. Square and add: x29 + y24 = 1. That is an ellipse reaching 3 horizontally and 2 vertically. Different radii mean it cannot be a circle.
Problem 6
For x = t, y = t2 with 0 ≤ t ≤ 2, what part of y = x2 is traced?
- From (0,0) to (2,4)
- The whole parabola
- From (-2,4) to (2,4)
- The left half
- A single point
Worked answer
From (0,0) to (2,4)
Since x = t, the restriction on t is a restriction on x: 0 ≤ x ≤ 2. At t=0 the point is (0,0); at t=2 it is (2,4). So only the right-hand piece is drawn — the eliminated equation y=x2 describes far more than the parametric one does.
Problem 7
What is the graph of x = sin t, y = sin t?
- The segment of y=x from (-1,-1) to (1,1)
- The whole line y=x
- A circle
- A single point
- A sine wave
Worked answer
The segment of y=x from (-1,-1) to (1,1)
Since both coordinates are the same, every point satisfies y = x. But sin t never leaves [-1,1], so the point only ever occupies that stretch. The result is a segment, traced back and forth forever — the full line is not reached.
Problem 8
A particle has x = t, y = 4 - t2. At what positive time does it cross the x-axis?
- t = 2
- t = 4
- t = 1
- t = √2
- t = 16
Worked answer
t = 2
Crossing the x-axis means y = 0. 4 - t2 = 0, so t2 = 4 and t = ± 2. The positive time is t = 2.